A manuscript was submitted describing a general criterion and two applications of it. The criterion allows one to distinguish between rings with the same "external action." Rings are algebraic structures with addition and multiplication satisfying certain rules. For example, the ordinary decimal numbers form a ring, as do the fractions, as do the integers, as do the systems of integer remainders obtained when dividing by a fixed positive integer. There are also rings constructed in more complicated ways, by taking matrices (arrays of numbers) for example. The study of rings is a major part of modern algebra. The "external actions" of rings distinguished by our new criterion are difficult to describe in nontechnical terms. Essentially, the algebraic properties of a ring can be viewed "from the outside," using structures that arise from the ring. We say that two rings have the same external action if the two rings lead to the same "outside" view. Rings with the same outside view must have the same "characteristic," which is zero for such rings as the decimal numbers or fractions, but is the integer d for the ring of integer remainders after division by d. All rings with the same characteristic are known to have the same external action if the characteristic is a prime number or a product of distinct primes. An application of the general criterion shows that there are infinitely many different external actions corresponding to each other possible ring characteristic. Another application shows that a ring need not have the same external action as its opposite or dual ring, obtained by reversing the factors in multiplications. The findings of this study are theorems of pure mathematics. The main purpose of such studies is to better understand the mathematical tools that have been successfully applied to many problems of science and engineering. There are numerous historical examples of advances in science and engineering. There are numerous historical examples of advances in science and technology, often unpredictable, resulting from such improvements in theoretical understanding.